The Eztrapolation Methods for Computing Supersingular Integral on Interval
Singular integral equations, including hypersingular and supersingular integral equations, are presently encountered in a wide range of nonlinear mathematical models. Such as acoustics, fluid mechanics, elasticity and fracture mechanics.The supersingular integral usually appears in many boundary elemant methods 1 and is not studied as widily as the hypersinguar integrqal, due to the singular kernel the convergence rate is not satisfy as the usual Riemann integral. Integrals with kernels beyond hypersingular have not been extensively studied. Du 2 firstly studied the composite Simpsons rule and showed the optimal global convergence rate is (h) Then, Wu and Sun 3 studied the superconvergence of trapezoidal rule and the (h2) superconvergence rate was obtained when the singular point is located at the middle point of each subinterval away from two endpoints. Recently, Zhang 4 et.al discussed the superconvergence phenomenon of the composite Simpsons rule and also the (h2) rate was obtained for those superconvergence points away from the endpoints.Extrapolation methods as an accelerating convergence technique has been applied to many fields in computational mathematics. The most famous one is Richardson extrapolation with the error function as T(h)-a0=a1h2+a2h4+…, where T(0)=a0 and aj are constant independent of h. Richardson eliminated the term h2 by combining the approximations with two different meshes. In each case he used pairs of approximation to eliminate h2.a process he named “ h2-extrapolation to improve numerical solutions of integral equation and ordinary differential equation. Then in the paper of Li 5 et.al, the trapezoidal rule for computation hypersingular integral by extrapolation methods was given. However, to our knowledge, no attempt has been made to apply extrapolation technique to accelerate convergence for the computation of supersingular integral.As we known, the composite trapezoidal rule for the computing the supersingular integral does not converge in general, as the convergence rate is two order lower than the Riemann integrals. While the superconvergence rate for supersingular is O(h2)the same as the Riemann integrals. In this paper we focus on the asymptotic error expansion of the composite trapezoidal rule for the computation of supersingular integrals. Based on the asymptotic error expansion En(f)=l-1∑i=1 ai(τ)hi-1+O(h1-2) where ai(τ)are functions independent of h, and τthe local coordinate of the singular point. We suggest an extrapolation algorithm. For a given τ, a series of Sjis selected to approximate the singular point s accompanied by the refinement of the meshes. Moreover, by means of the extrapolation technique we not only obtain an approximation with higher order accuracy, but also get a posteriori error estimate. At last, some numerical results are also reported to confirm the theoretical results and show the efficiency of the algorithms.
J.li D.H.Yu
LSEC,Institute of Computational Mathematics andScientific/Engineering Computing,Academy ofMathematics and SystemsScience,CAS,Beijing 100080,P.R.China
国际会议
南京
英文
1-2
2009-10-18（万方平台首次上网日期，不代表论文的发表时间）